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Theorem canthwdom 9617
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 9169, equivalent to canth 7385). (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
canthwdom ¬ 𝒫 𝐴* 𝐴

Proof of Theorem canthwdom
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0elpw 5362 . . . . 5 ∅ ∈ 𝒫 𝐴
2 ne0i 4347 . . . . 5 (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅)
31, 2mp1i 13 . . . 4 (𝒫 𝐴* 𝐴 → 𝒫 𝐴 ≠ ∅)
4 brwdomn0 9607 . . . 4 (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴))
53, 4syl 17 . . 3 (𝒫 𝐴* 𝐴 → (𝒫 𝐴* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴))
65ibi 267 . 2 (𝒫 𝐴* 𝐴 → ∃𝑓 𝑓:𝐴onto→𝒫 𝐴)
7 relwdom 9604 . . . . 5 Rel ≼*
87brrelex2i 5746 . . . 4 (𝒫 𝐴* 𝐴𝐴 ∈ V)
9 foeq2 6818 . . . . . . 7 (𝑥 = 𝐴 → (𝑓:𝑥onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝑥))
10 pweq 4619 . . . . . . . 8 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
11 foeq3 6819 . . . . . . . 8 (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
1210, 11syl 17 . . . . . . 7 (𝑥 = 𝐴 → (𝑓:𝐴onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
139, 12bitrd 279 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝑥onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
1413notbid 318 . . . . 5 (𝑥 = 𝐴 → (¬ 𝑓:𝑥onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴onto→𝒫 𝐴))
15 vex 3482 . . . . . 6 𝑥 ∈ V
1615canth 7385 . . . . 5 ¬ 𝑓:𝑥onto→𝒫 𝑥
1714, 16vtoclg 3554 . . . 4 (𝐴 ∈ V → ¬ 𝑓:𝐴onto→𝒫 𝐴)
188, 17syl 17 . . 3 (𝒫 𝐴* 𝐴 → ¬ 𝑓:𝐴onto→𝒫 𝐴)
1918nexdv 1934 . 2 (𝒫 𝐴* 𝐴 → ¬ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴)
206, 19pm2.65i 194 1 ¬ 𝒫 𝐴* 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1537  wex 1776  wcel 2106  wne 2938  Vcvv 3478  c0 4339  𝒫 cpw 4605   class class class wbr 5148  ontowfo 6561  * cwdom 9602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-wdom 9603
This theorem is referenced by:  pwdjudom  10253
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